Question: Solve for $x$: $3^{2x} = \sqrt{27}$. Express your answer as a common fraction.
Solution: Because $\sqrt{27} = 27^{\frac{1}{2}} = (3^3)^\frac{1}{2} = 3^{\frac{3}{2}}$, we have $3^{2x}=3^{\frac{3}{2}}$.  This gives us $2x=\frac{3}{2}$, so $x=\boxed{\frac{3}{4}}$.